Here is a proposition quoted from the book written by do Carmo. I don't understand the part underlined with red. Or equivalently (I guess), I don't know how it works to make sure that the given field $N$ is, on every coordinate neighborhood, compatible with the family of connected coordinate neighborhood. By compatibility, I mean
$$N=\frac{X_u\wedge X_v}{|X_u\wedge X_v|},$$
where $X$ is a parametrization of a coordinate neighborhood in the family. Thank you.
One may not know the definition of the change of coordinates. It is simply the transition map between two overlapping surface patches. But more information is fine.
Best Answer
Interchanging $u$ and $v$ means the following: You have a chart $x : U \to V \cap S, (u,v) \mapsto x(u,v)$. Now consider the homeomorphism $\tau : \mathbb R^2 \to \mathbb R^2, \tau(u,v) = (v,u)$. Note $\tau \circ \tau = id$. Then $x' : \tau(U) \stackrel{\tau}{\rightarrow} U \stackrel{x}{\rightarrow} V \cap S$ is a new chart in which $(u,v) \mapsto x(v,u)$. Both charts parametrize the same coordinate neigborhood $V \cap S$.
The idea is to start with any family of charts having connected coordinate neighborhoods. Connectedness assures that for all $(u,v)$ either $N=\frac{x_u\wedge x_v}{|x_u\wedge x_v|}$ or $N=-\frac{x_u\wedge x_v}{|x_u\wedge x_v|}$. In the second case use the above method to replace the original chart by the "switched" chart. Thus the field $N$ is used to find a new family of charts which is "normalized" by the condition $N=\frac{x_u\wedge x_v}{|x_u\wedge x_v|}$.